Optimal. Leaf size=284 \[ -\frac {3 \left (4 a^2-14 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 d (a-b)^{7/2}}+\frac {3 \left (4 a^2+14 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 d (a+b)^{7/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)+b \left (a^2+9 b^2\right )\right )}{16 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}-\frac {3 b \left (2 a^4-7 a^2 b^2-15 b^4\right )}{16 d \left (a^2-b^2\right )^3 \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.52, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2668, 741, 823, 829, 827, 1166, 206} \[ -\frac {3 b \left (-7 a^2 b^2+2 a^4-15 b^4\right )}{16 d \left (a^2-b^2\right )^3 \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (4 a^2-14 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 d (a-b)^{7/2}}+\frac {3 \left (4 a^2+14 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 d (a+b)^{7/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)+b \left (a^2+9 b^2\right )\right )}{16 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 741
Rule 823
Rule 827
Rule 829
Rule 1166
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {1}{(a+x)^{3/2} \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\frac {3}{2} \left (2 a^2-3 b^2\right )+\frac {7 a x}{2}}{(a+x)^{3/2} \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 \left (a^2-b^2\right ) d}\\ &=-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (a^2+9 b^2\right )+2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {b \operatorname {Subst}\left (\int \frac {-\frac {3}{4} \left (4 a^4-9 a^2 b^2+15 b^4\right )-\frac {3}{2} a \left (3 a^2-8 b^2\right ) x}{(a+x)^{3/2} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ &=-\frac {3 b \left (2 a^4-7 a^2 b^2-15 b^4\right )}{16 \left (a^2-b^2\right )^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (a^2+9 b^2\right )+2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}+\frac {b \operatorname {Subst}\left (\int \frac {\frac {3}{4} a \left (4 a^4-15 a^2 b^2+31 b^4\right )+\frac {3}{4} \left (2 a^4-7 a^2 b^2-15 b^4\right ) x}{\sqrt {a+x} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^3 d}\\ &=-\frac {3 b \left (2 a^4-7 a^2 b^2-15 b^4\right )}{16 \left (a^2-b^2\right )^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (a^2+9 b^2\right )+2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}+\frac {b \operatorname {Subst}\left (\int \frac {-\frac {3}{4} a \left (2 a^4-7 a^2 b^2-15 b^4\right )+\frac {3}{4} a \left (4 a^4-15 a^2 b^2+31 b^4\right )+\frac {3}{4} \left (2 a^4-7 a^2 b^2-15 b^4\right ) x^2}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{4 \left (a^2-b^2\right )^3 d}\\ &=-\frac {3 b \left (2 a^4-7 a^2 b^2-15 b^4\right )}{16 \left (a^2-b^2\right )^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (a^2+9 b^2\right )+2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (3 \left (4 a^2-14 a b+15 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{32 (a-b)^3 d}+\frac {\left (3 \left (4 a^2+14 a b+15 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{32 (a+b)^3 d}\\ &=-\frac {3 \left (4 a^2-14 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 (a-b)^{7/2} d}+\frac {3 \left (4 a^2+14 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 (a+b)^{7/2} d}-\frac {3 b \left (2 a^4-7 a^2 b^2-15 b^4\right )}{16 \left (a^2-b^2\right )^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (a^2+9 b^2\right )+2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 2.24, size = 324, normalized size = 1.14 \[ \frac {3 a \sqrt {a-b} \sqrt {a+b} \left (3 a^2-8 b^2\right ) \sqrt {a+b \sin (c+d x)} \left (\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )-\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )\right )-(a-b) (a+b) \sec ^2(c+d x) \left (2 a \left (3 a^2-8 b^2\right ) \sin (c+d x)+b \left (a^2+9 b^2\right )\right )+\frac {3}{2} \left (2 a^4-7 a^2 b^2-15 b^4\right ) \left ((a+b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \sin (c+d x)}{a-b}\right )+(b-a) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \sin (c+d x)}{a+b}\right )\right )-4 (a-b)^2 (a+b)^2 \sec ^4(c+d x) (a \sin (c+d x)-b)}{16 d \left (a^2-b^2\right )^2 \left (b^2-a^2\right ) \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{5}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{5}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.10, size = 649, normalized size = 2.29 \[ \frac {2 b^{5}}{d \left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {a +b \sin \left (d x +c \right )}}-\frac {3 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} a}{16 d \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {13 b^{2} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{32 d \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {3 b \sqrt {a +b \sin \left (d x +c \right )}\, a^{2}}{16 d \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}-\frac {21 b^{2} \sqrt {a +b \sin \left (d x +c \right )}\, a}{32 d \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {15 b^{3} \sqrt {a +b \sin \left (d x +c \right )}}{32 d \left (a -b \right )^{3} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a^{2}}{8 d \left (a -b \right )^{3} \sqrt {-a +b}}-\frac {21 b \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a}{16 d \left (a -b \right )^{3} \sqrt {-a +b}}+\frac {45 b^{2} \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{32 d \left (a -b \right )^{3} \sqrt {-a +b}}-\frac {3 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} a}{16 d \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}-\frac {13 b^{2} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{32 d \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {3 b \sqrt {a +b \sin \left (d x +c \right )}\, a^{2}}{16 d \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {21 b^{2} \sqrt {a +b \sin \left (d x +c \right )}\, a}{32 d \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {15 b^{3} \sqrt {a +b \sin \left (d x +c \right )}}{32 d \left (a +b \right )^{3} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {3 \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a^{2}}{8 d \left (a +b \right )^{\frac {7}{2}}}+\frac {21 b \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a}{16 d \left (a +b \right )^{\frac {7}{2}}}+\frac {45 b^{2} \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{32 d \left (a +b \right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^5\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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