Optimal. Leaf size=747 \[ -\frac {b^{5/3} \left (4 a^2-3 a^{4/3} b^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (4 a^{8/3}-9 a^2 b^{2/3}+8 a^{2/3} b^2-3 b^{8/3}\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )} \]
[Out]
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Rubi [A]
time = 0.68, antiderivative size = 747, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3302, 2099,
1868, 1874, 31, 648, 631, 210, 642, 1885, 266} \begin {gather*} -\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )\right )}{3 a d \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^3}-\frac {b^{5/3} \left (-3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} d \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (8 a^{2/3} b^2+4 a^{8/3}-9 a^2 b^{2/3}-3 b^{8/3}\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} d \left (a^2-b^2\right )^3}-\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} d \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} d \left (a^2-b^2\right )^3}+\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} d \left (a^2-b^2\right )^2}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} d \left (a^2-b^2\right )^3}+\frac {1}{4 d (a+b)^2 (1-\sin (c+d x))}-\frac {1}{4 d (a-b)^2 (\sin (c+d x)+1)}-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac {(a-7 b) \log (\sin (c+d x)+1)}{4 d (a-b)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1868
Rule 1874
Rule 1885
Rule 2099
Rule 3302
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{4 (a+b)^2 (-1+x)^2}+\frac {-a-7 b}{4 (a+b)^3 (-1+x)}+\frac {1}{4 (a-b)^2 (1+x)^2}+\frac {a-7 b}{4 (a-b)^3 (1+x)}+\frac {b^2 \left (2 a^2+b^2-3 a b x+\left (a^2+2 b^2\right ) x^2\right )}{\left (a^2-b^2\right )^2 \left (a+b x^3\right )^2}+\frac {b^2 \left (4 a \left (a^2+2 b^2\right )-3 b \left (3 a^2+b^2\right ) x+2 a \left (a^2+5 b^2\right ) x^2\right )}{\left (a^2-b^2\right )^3 \left (a+b x^3\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^2 \text {Subst}\left (\int \frac {4 a \left (a^2+2 b^2\right )-3 b \left (3 a^2+b^2\right ) x+2 a \left (a^2+5 b^2\right ) x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {b^2 \text {Subst}\left (\int \frac {2 a^2+b^2-3 a b x+\left (a^2+2 b^2\right ) x^2}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {b^2 \text {Subst}\left (\int \frac {4 a \left (a^2+2 b^2\right )-3 b \left (3 a^2+b^2\right ) x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^3 d}-\frac {b^2 \text {Subst}\left (\int \frac {-2 \left (2 a^2+b^2\right )+3 a b x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a \left (a^2-b^2\right )^2 d}+\frac {\left (2 a b^2 \left (a^2+5 b^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {b^{5/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (-3 \sqrt [3]{a} b \left (3 a^2+b^2\right )+8 a \sqrt [3]{b} \left (a^2+2 b^2\right )\right )+\sqrt [3]{b} \left (-3 \sqrt [3]{a} b \left (3 a^2+b^2\right )-4 a \sqrt [3]{b} \left (a^2+2 b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^3 d}-\frac {b^{5/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (3 a^{4/3} b-4 \sqrt [3]{b} \left (2 a^2+b^2\right )\right )+\sqrt [3]{b} \left (3 a^{4/3} b+2 \sqrt [3]{b} \left (2 a^2+b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (b^2 \left (4 a^2-3 a^{4/3} b^{2/3}+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{6 a^{4/3} \left (a^2-b^2\right )^2 d}-\frac {\left (b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (4 a^{8/3}-9 a^2 b^{2/3}+8 a^{2/3} b^2-3 b^{8/3}\right )\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{2 \left (a^2-b^2\right )^3 d}-\frac {\left (b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (b^{5/3} \left (4 a^2-3 a^{4/3} b^{2/3}+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {\left (b^{5/3} \left (4 a^{8/3}-9 a^2 b^{2/3}+8 a^{2/3} b^2-3 b^{8/3}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} \left (a^2-b^2\right )^3 d}\\ &=-\frac {b^{5/3} \left (4 a^2-3 a^{4/3} b^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (4 a^{8/3}-9 a^2 b^{2/3}+8 a^{2/3} b^2-3 b^{8/3}\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 6.30, size = 657, normalized size = 0.88 \begin {gather*} \frac {-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3}+\frac {4 \sqrt [3]{a} b^{5/3} \left (a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3}-\frac {2 \sqrt [3]{a} \left (a^2+2 b^2\right ) \left (2 \sqrt {3} b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )+b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )\right )}{3 \left (a^2-b^2\right )^3}+\frac {\left (2+\frac {b^2}{a^2}\right ) \left (2 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )-\sqrt [3]{a} \left (2 \sqrt {3} b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )+b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )\right )\right )}{9 \left (a^2-b^2\right )^2}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3}+\frac {1}{4 (a+b)^2 (1-\sin (c+d x))}-\frac {3 b^3 \left (3 a^2+b^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{2 a \left (a^2-b^2\right )^3}-\frac {3 b^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{2 a \left (a^2-b^2\right )^2}-\frac {1}{4 (a-b)^2 (1+\sin (c+d x))}-\frac {b \left (a^2+2 b^2\right )}{3 \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}+\frac {a b^2 \left (2+\frac {b^2}{a^2}\right ) \sin (c+d x)}{3 \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 2.84, size = 483, normalized size = 0.65
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} \left (\frac {\left (-a^{2} b +b^{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )+\frac {\left (2 a^{4}-a^{2} b^{2}-b^{4}\right ) \sin \left (d x +c \right )}{3 a}-\frac {a^{4}+a^{2} b^{2}-2 b^{4}}{3 b}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (8 a^{4}+11 a^{2} b^{2}-b^{4}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (-15 a^{3} b -3 a \,b^{3}\right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {2 \left (3 a^{4}+15 a^{2} b^{2}\right ) \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{9 b}}{a}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {1}{4 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -7 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{3}}}{d}\) | \(483\) |
default | \(\frac {\frac {b^{2} \left (\frac {\left (-a^{2} b +b^{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )+\frac {\left (2 a^{4}-a^{2} b^{2}-b^{4}\right ) \sin \left (d x +c \right )}{3 a}-\frac {a^{4}+a^{2} b^{2}-2 b^{4}}{3 b}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (8 a^{4}+11 a^{2} b^{2}-b^{4}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (-15 a^{3} b -3 a \,b^{3}\right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {2 \left (3 a^{4}+15 a^{2} b^{2}\right ) \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{9 b}}{a}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {1}{4 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -7 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{3}}}{d}\) | \(483\) |
risch | \(\text {Expression too large to display}\) | \(1831\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 788, normalized size = 1.05 \begin {gather*} -\frac {\frac {8 \, \sqrt {3} {\left (5 \, a^{3} b^{2} {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2\right )} - a^{2} b^{3} {\left (11 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {10 \, a}{b}\right )} - 2 \, a^{4} b {\left (4 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {a}{b}\right )} + 3 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{5}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (a^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{5} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {4 \, {\left (a^{2} b^{3} {\left (30 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 11\right )} + 2 \, a^{4} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4\right )} - 15 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 3 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}} + b^{5}\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{5} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {8 \, {\left (a^{2} b^{3} {\left (15 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 11\right )} + a^{4} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 8\right )} + 15 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}} - b^{5}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{5} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {9 \, {\left (a - 7 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {9 \, {\left (a + 7 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {6 \, {\left (3 \, {\left (a^{3} b + 3 \, a b^{3}\right )} \sin \left (d x + c\right )^{4} - 8 \, a^{3} b - 4 \, a b^{3} - 2 \, {\left (5 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{3} + 2 \, {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{2} + {\left (3 \, a^{4} + 7 \, a^{2} b^{2} + 2 \, b^{4}\right )} \sin \left (d x + c\right )\right )}}{a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4} - {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{3} - {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{2}}}{36 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 6.24, size = 15989, normalized size = 21.40 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 790, normalized size = 1.06 \begin {gather*} \frac {\frac {8 \, {\left (15 \, a^{10} b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 42 \, a^{8} b^{6} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 36 \, a^{6} b^{8} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 6 \, a^{4} b^{10} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 3 \, a^{2} b^{12} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 8 \, a^{11} b^{3} + 13 \, a^{9} b^{5} + 10 \, a^{7} b^{7} - 28 \, a^{5} b^{9} + 14 \, a^{3} b^{11} - a b^{13}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a^{15} b - 6 \, a^{13} b^{3} + 15 \, a^{11} b^{5} - 20 \, a^{9} b^{7} + 15 \, a^{7} b^{9} - 6 \, a^{5} b^{11} + a^{3} b^{13}} + \frac {8 \, {\left (3 \, {\left (5 \, \sqrt {3} a^{3} b + \sqrt {3} a b^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} + {\left (8 \, \sqrt {3} a^{4} b + 11 \, \sqrt {3} a^{2} b^{3} - \sqrt {3} b^{5}\right )} \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} - \frac {4 \, {\left (3 \, {\left (5 \, a^{3} b + a b^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} - {\left (8 \, a^{4} b + 11 \, a^{2} b^{3} - b^{5}\right )} \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} + \frac {24 \, {\left (a^{3} b + 5 \, a b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {9 \, {\left (a - 7 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {9 \, {\left (a + 7 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {6 \, {\left (3 \, a^{3} b \sin \left (d x + c\right )^{4} + 9 \, a b^{3} \sin \left (d x + c\right )^{4} - 10 \, a^{2} b^{2} \sin \left (d x + c\right )^{3} - 2 \, b^{4} \sin \left (d x + c\right )^{3} + 2 \, a^{3} b \sin \left (d x + c\right )^{2} - 2 \, a b^{3} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + 7 \, a^{2} b^{2} \sin \left (d x + c\right ) + 2 \, b^{4} \sin \left (d x + c\right ) - 8 \, a^{3} b - 4 \, a b^{3}\right )}}{{\left (b \sin \left (d x + c\right )^{5} - b \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - a\right )} {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}}}{36 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.75, size = 1605, normalized size = 2.15 \begin {gather*} \frac {\sum _{k=1}^3\ln \left (\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )\,\left (-\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )\,\left (\frac {-\frac {63\,a^{10}\,b^4}{2}+\frac {4153\,a^8\,b^6}{12}+325\,a^6\,b^8-\frac {1017\,a^4\,b^{10}}{4}+\frac {32\,a^2\,b^{12}}{3}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )\,\left (\frac {\frac {27\,a^{13}\,b^3}{2}-239\,a^{11}\,b^5+188\,a^9\,b^7+303\,a^7\,b^9-\frac {563\,a^5\,b^{11}}{2}+16\,a^3\,b^{13}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )\,\left (\frac {180\,a^{14}\,b^4-684\,a^{12}\,b^6+936\,a^{10}\,b^8-504\,a^8\,b^{10}+36\,a^6\,b^{12}+36\,a^4\,b^{14}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\frac {\sin \left (c+d\,x\right )\,\left (5832\,a^{15}\,b^3-5832\,a^{13}\,b^5-34992\,a^{11}\,b^7+81648\,a^9\,b^9-64152\,a^7\,b^{11}+17496\,a^5\,b^{13}\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )-\frac {\sin \left (c+d\,x\right )\,\left (2916\,a^{12}\,b^4+50004\,a^{10}\,b^6-96660\,a^8\,b^8+30780\,a^6\,b^{10}+13824\,a^4\,b^{12}-864\,a^2\,b^{14}\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )-\frac {\sin \left (c+d\,x\right )\,\left (10449\,a^9\,b^5+31542\,a^7\,b^7-68247\,a^5\,b^9+7200\,a^3\,b^{11}-384\,a\,b^{13}\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )+\frac {-20\,a^7\,b^5+\frac {847\,a^5\,b^7}{3}+\frac {2173\,a^3\,b^9}{27}+\frac {32\,a\,b^{11}}{27}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\frac {\sin \left (c+d\,x\right )\,\left (-9234\,a^6\,b^6-29860\,a^4\,b^8+4758\,a^2\,b^{10}+64\,b^{12}\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )+\frac {\frac {10\,a^4\,b^6}{3}+\frac {122\,a^2\,b^8}{27}+\frac {28\,b^{10}}{27}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\frac {\sin \left (c+d\,x\right )\,\left (2568\,a^3\,b^7+1080\,a\,b^9\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )\,\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )}{d}+\frac {\frac {b\,{\sin \left (c+d\,x\right )}^2}{3\,\left (a^2-b^2\right )}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {a^2\,b}{2}+\frac {3\,b^3}{2}\right )}{a^4-2\,a^2\,b^2+b^4}-\frac {2\,b\,\left (2\,a^2+b^2\right )}{3\,{\left (a^2-b^2\right )}^2}+\frac {\sin \left (c+d\,x\right )\,\left (3\,a^4+7\,a^2\,b^2+2\,b^4\right )}{6\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {5\,a^2\,b^2}{3}+\frac {b^4}{3}\right )}{a\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (-b\,{\sin \left (c+d\,x\right )}^5+b\,{\sin \left (c+d\,x\right )}^3-a\,{\sin \left (c+d\,x\right )}^2+a\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (a+7\,b\right )}{d\,\left (4\,a^3+12\,a^2\,b+12\,a\,b^2+4\,b^3\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-7\,b\right )}{d\,\left (4\,a^3-12\,a^2\,b+12\,a\,b^2-4\,b^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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