Optimal. Leaf size=150 \[ -\frac {2 \left (a^2-b^2\right )^2}{b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {8 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}{b^5 d}+\frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac {2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2747, 711}
\begin {gather*} \frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}-\frac {8 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}{b^5 d}-\frac {2 \left (a^2-b^2\right )^2}{b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{5/2}}{5 b^5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rule 2747
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{(a+x)^{3/2}} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {\left (a^2-b^2\right )^2}{(a+x)^{3/2}}-\frac {4 \left (a^3-a b^2\right )}{\sqrt {a+x}}+2 \left (3 a^2-b^2\right ) \sqrt {a+x}-4 a (a+x)^{3/2}+(a+x)^{5/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac {2 \left (a^2-b^2\right )^2}{b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {8 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}{b^5 d}+\frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac {2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d}\\ \end {align*}
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Mathematica [A]
time = 0.69, size = 165, normalized size = 1.10 \begin {gather*} \frac {-3072 a^4+4672 a^2 b^2-1075 b^4-4 \left (48 a^2 b^2-55 b^4\right ) \cos (2 (c+d x))+15 b^4 \cos (4 (c+d x))-1536 a^3 b \sin (c+d x)+2096 a b^3 \sin (c+d x)+2232 a^4 \sqrt {1+\frac {b \sin (c+d x)}{a}}-2644 a^2 b^2 \sqrt {1+\frac {b \sin (c+d x)}{a}}+48 a b^3 \sin (3 (c+d x))}{420 b^5 d \sqrt {a+b \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.00, size = 116, normalized size = 0.77
method | result | size |
default | \(\frac {\frac {16 a \,b^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}+\frac {2 \left (-192 b \,a^{3}+256 a \,b^{3}\right ) \sin \left (d x +c \right )}{105}+\frac {2 b^{4} \left (\cos ^{4}\left (d x +c \right )\right )}{7}+\frac {2 \left (-48 a^{2} b^{2}+40 b^{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right )}{105}-\frac {256 a^{4}}{35}+\frac {1216 a^{2} b^{2}}{105}-\frac {64 b^{4}}{21}}{b^{5} \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 124, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (\frac {15 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 84 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 70 \, {\left (3 \, a^{2} - b^{2}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - 420 \, {\left (a^{3} - a b^{2}\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{b^{4}} - \frac {105 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}}{\sqrt {b \sin \left (d x + c\right ) + a} b^{4}}\right )}}{105 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 125, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (15 \, b^{4} \cos \left (d x + c\right )^{4} - 384 \, a^{4} + 608 \, a^{2} b^{2} - 160 \, b^{4} - 8 \, {\left (6 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, a b^{3} \cos \left (d x + c\right )^{2} - 24 \, a^{3} b + 32 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{105 \, {\left (b^{6} d \sin \left (d x + c\right ) + a b^{5} d\right )}} \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 [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^5}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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