Optimal. Leaf size=135 \[ \frac {2 \sqrt {b} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{d^{3/2}}+\frac {2 \sqrt {b} \sqrt {\pi } C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{d^{3/2}}-\frac {2 \sin ^2(a+b x)}{d \sqrt {c+d x}} \]
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Rubi [A]
time = 0.17, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3394, 12, 3387,
3386, 3432, 3385, 3433} \begin {gather*} \frac {2 \sqrt {\pi } \sqrt {b} \sin \left (2 a-\frac {2 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {\pi } \sqrt {d}}\right )}{d^{3/2}}+\frac {2 \sqrt {\pi } \sqrt {b} \cos \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{d^{3/2}}-\frac {2 \sin ^2(a+b x)}{d \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3394
Rule 3432
Rule 3433
Rubi steps
\begin {align*} \int \frac {\sin ^2(a+b x)}{(c+d x)^{3/2}} \, dx &=-\frac {2 \sin ^2(a+b x)}{d \sqrt {c+d x}}+\frac {(4 b) \int \frac {\sin (2 a+2 b x)}{2 \sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 \sin ^2(a+b x)}{d \sqrt {c+d x}}+\frac {(2 b) \int \frac {\sin (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 \sin ^2(a+b x)}{d \sqrt {c+d x}}+\frac {\left (2 b \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{d}+\frac {\left (2 b \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 \sin ^2(a+b x)}{d \sqrt {c+d x}}+\frac {\left (4 b \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}+\frac {\left (4 b \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {2 \sqrt {b} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{d^{3/2}}+\frac {2 \sqrt {b} \sqrt {\pi } C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{d^{3/2}}-\frac {2 \sin ^2(a+b x)}{d \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 149, normalized size = 1.10 \begin {gather*} \frac {-1+\cos (2 (a+b x))+2 \sqrt {\frac {b}{d}} \sqrt {\pi } \sqrt {c+d x} \cos \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )+2 \sqrt {\frac {b}{d}} \sqrt {\pi } \sqrt {c+d x} C\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{d \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 145, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {-\frac {1}{\sqrt {d x +c}}+\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 c b}{d}\right )}{\sqrt {d x +c}}+\frac {2 b \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 c b}{d}\right ) \mathrm {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 d a -2 c b}{d}\right ) \FresnelC \left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}}{d}\) | \(145\) |
default | \(\frac {-\frac {1}{\sqrt {d x +c}}+\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 c b}{d}\right )}{\sqrt {d x +c}}+\frac {2 b \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 c b}{d}\right ) \mathrm {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 d a -2 c b}{d}\right ) \FresnelC \left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}}{d}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.60, size = 135, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, \frac {2 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -\frac {2 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, \frac {2 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -\frac {2 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sqrt {\frac {{\left (d x + c\right )} b}{d}} + 8}{8 \, \sqrt {d x + c} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 138, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left ({\left (\pi d x + \pi c\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + {\left (\pi d x + \pi c\right )} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \sqrt {d x + c} {\left (\cos \left (b x + a\right )^{2} - 1\right )}\right )}}{d^{2} x + c d} \end {gather*}
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 \begin {gather*} \int \frac {\sin ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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