Optimal. Leaf size=207 \[ -\frac {\left (12 a^2-18 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 d (a-b)^{3/2}}+\frac {\left (12 a^2+18 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 d (a+b)^{3/2}}-\frac {\sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (a b-\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{16 d \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 d} \]
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Rubi [A] time = 0.32, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2668, 737, 823, 827, 1166, 206} \[ -\frac {\left (12 a^2-18 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 d (a-b)^{3/2}}+\frac {\left (12 a^2+18 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 d (a+b)^{3/2}}-\frac {\sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (a b-\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{16 d \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{4 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 737
Rule 823
Rule 827
Rule 1166
Rule 2668
Rubi steps
\begin {align*} \int \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {\sqrt {a+x}}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \tan (c+d x)}{4 d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {-3 a-\frac {5 x}{2}}{\sqrt {a+x} \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=-\frac {\sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (a b-\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right ) d}+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \tan (c+d x)}{4 d}+\frac {b \operatorname {Subst}\left (\int \frac {\frac {1}{4} a \left (12 a^2-13 b^2\right )+\frac {1}{4} \left (6 a^2-5 b^2\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 ) d}\\ &=-\frac {\sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (a b-\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right ) d}+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \tan (c+d x)}{4 d}+\frac {b \operatorname {Subst}\left (\int \frac {\frac {1}{4} a \left (12 a^2-13 b^2\right )-\frac {1}{4} a \left (6 a^2-5 b^2\right )+\frac {1}{4} \left (6 a^2-5 b^2\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 ) d}\\ &=-\frac {\sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (a b-\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right ) d}+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \tan (c+d x)}{4 d}-\frac {\left (12 a^2-18 a b+5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{32 (a-b) d}+\frac {\left (12 a^2+18 a b+5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{32 (a+b) d}\\ &=-\frac {\left (12 a^2-18 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 (a-b)^{3/2} d}+\frac {\left (12 a^2+18 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 (a+b)^{3/2} d}-\frac {\sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (a b-\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right ) d}+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 1.52, size = 224, normalized size = 1.08 \[ \frac {-\sqrt {a-b} (a+b)^2 \left (12 a^2-18 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )+(a-b)^2 \sqrt {a+b} \left (12 a^2+18 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )+\frac {1}{2} \left (a^2-b^2\right ) \sec ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (22 a^2-21 b^2\right ) \sin (c+d x)+6 a^2 \sin (3 (c+d x))-2 a b \cos (2 (c+d x))-2 a b-5 b^2 \sin (3 (c+d x))\right )}{32 d \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{5}, x\right ) \]
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 \[ \int \sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.88, size = 509, normalized size = 2.46 \[ -\frac {3 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} a}{16 d \left (b \sin \left (d x +c \right )+b \right )^{2} \left (a -b \right )}+\frac {5 b^{2} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{32 d \left (b \sin \left (d x +c \right )+b \right )^{2} \left (a -b \right )}+\frac {3 b \sqrt {a +b \sin \left (d x +c \right )}\, a}{16 d \left (b \sin \left (d x +c \right )+b \right )^{2}}-\frac {7 b^{2} \sqrt {a +b \sin \left (d x +c \right )}}{32 d \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a^{2}}{8 d \left (a -b \right ) \sqrt {-a +b}}-\frac {9 \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a b}{16 d \left (a -b \right ) \sqrt {-a +b}}+\frac {5 \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) b^{2}}{32 d \left (a -b \right ) \sqrt {-a +b}}-\frac {3 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} a}{16 d \left (b \sin \left (d x +c \right )-b \right )^{2} \left (a +b \right )}-\frac {5 b^{2} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{32 d \left (b \sin \left (d x +c \right )-b \right )^{2} \left (a +b \right )}+\frac {3 b \sqrt {a +b \sin \left (d x +c \right )}\, a}{16 d \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {7 b^{2} \sqrt {a +b \sin \left (d x +c \right )}}{32 d \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {3 \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a^{2}}{8 d \left (a +b \right )^{\frac {3}{2}}}+\frac {9 \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a b}{16 d \left (a +b \right )^{\frac {3}{2}}}+\frac {5 \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) b^{2}}{32 d \left (a +b \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {a+b\,\sin \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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